Gary Greergreerg@limestone.on.cahttp://napaneedss.limestone.on.ca/greer

Materials Available Onlinehttp://napaneedss.limestone.on.ca/greer/teachers/cm/cmap.html

Concept Maps in Mathematics Education

Gary Greergreerg@limestone.on.ca

Cape Breton UniversityMemorial University of Newfoundland

Part 1: What are Concept Maps?

Concept Maps?

What are Propositions?• concept maps constructed of propositions

(two concepts connected by a short linking phrase) which are the unit of meaning (perhaps analogous to sentence)

• the brief linking phrase describes the relationship between the two concepts

• anyone can draw lines between words, but connecting these with a phrase creates the window into understanding

A Mathematical Example

• here are two common ideas from secondary school math

• what linking phrase should we use?• “is a”? “represents”? • should the arrow go the other way?• Is there some common root that would better

define the relationship?

Concept Maps: Structure• the classic Novak map is hierarchical but

this may not be necessary in all situations

Sample Concept Map: Numbers

• since concept maps are idiosyncratic, each person may construct their own valid representation

• what does this say about assessment?

Theoretical Basis

• concept mapping is grounded in Ausubel’s assimilation theory– new knowledge is assimilated into previously held

knowledge structures• use is supported by the idea of cognitive off-

loading• also supported by constructivist approaches to

learning and by the idea of a zone of proximal development

Software• abundance of applications• creative, colorful, links to resources• students may be more willing to improve it• easy to edit• can create web pages• support collaborative learning• Inspiration/Kidspiration - widely used, cost• CMapTools - free, powerful, collaborative• SmartIdeas - provincially licensed

SmartIdeas• experiment…typing anywhere will create a

concept• clicking once on a concept enables the blue

linking arrow– click, hold and drag to link

• clicking twice should enable link editing• to change the appearance, click once on the

concept to select it and click on the desired design at the left

• right clicking presents you with menus• try creating a concept map explaining adding and

subtracting integers

Part 2: Analysis of Research into the Effectiveness of Concept Maps in Mathematics Education

My Study

• my final project of M.Ed. (I.T.) investigated whether maps are effective in mathematics

education as suggested in literature• concept maps developed by Novak in the 1980’s

and widely used in Science education• motivation:

– increasing concerns over achievement in Math– shift from procedural to conceptual learning with

increased emphasis on relationships, connectedness – availability of powerful concept mapping software– personal interest in using technology

Research Questions• Overall

– Will a review of literature provide supporting evidence that concept maps are effective tools in facilitating learning and assessing student knowledge in mathematics education?

• Specific Questions– Are concept maps valid and reliable

instruments for assessing mathematical knowledge?

– Can improvements in student learning be attributed to the use of concept maps?

The Study• an analysis of 11 available peer-reviewed

research papers was undertaken• these were analyzed with respect to validity and

reliability as well as links between the use of concept maps and learning

• 5 to 108 subjects, • 1 elementary, 3 secondary and 7 post-secondary• assessment varied between qualitative,

quantitative, holistic, and combinations of these• see Research Paper

Summary of Studies

qualitative3 maps over a semester

traditional exam & interview

Post-Secondary26McGowen and Tall (1999)

quantitativeone timeinterviewsPost-Secondary17Mwakapenda (2005)

holistictwicePost-Secondary19Roberts (1999)

holisticone timePost-Secondary28Williams (1998)

qualitativeone timeunsupervised examPost-Secondary5Zwaneveld (2000)

holisticone timeessay & traditional exam

Post-Secondary108Bolte (1999)

quantitativesemestervee diagram & collaboration

Post-Secondary16Afamasaga-Fuata’i (2004)

holisticone timehypermedia, interview & exam

Secondary20Flanagan (2002)

quantitativeone timetraditional examSecondary48Barolos (2002)

quantitativepre & post instruction

traditional examSecondary23

qualitativetwice over5 months

interview & problem solving

Elementary51Hasemann and Mansfield (1995)

Overall AssessmentType

InteractionDuration

Corroborating Measures

StudentLevel

nStudy

Findings: • generally correlate with other instruments• great variability, but there is some support for

assessment validity• great variability, but there is some support for

concept maps as reliable instruments• while learning was observed to have occurred,

there was insufficient effort to limit the effect of external factors – no control groups

• no clear empirical link was established between the use of concept maps and learning

• indirect evidence was supportive however

Findings: Participant Perspective

• researchers generally supportive of concept maps as effective assessment instruments

• specifically useful in identifying student misconceptions

• researchers indicated that maps were useful in facilitating student learning

• students indicated that concept maps were useful in promoting understanding

Summary of Analysis

moderategood (C), moderate (I),moderate (E)

McGowen and Tall (1999)

modestmoderate (C), low (I), modest (E)

Mwakapenda (2005)

modestmoderate (C), moderate (I), low (E)

questionableRoberts (1999)

modestmoderate (C), good (I), moderate (E)

Williams (1998)

lowmodest (C), moderate (I),low (E)

no clear linkZwaneveld (2000)

moderatemodest (C), good (I),good (E)

Bolte (1999)

lowmoderate (C), modest (I), modest (E)

no clear linkAfamasaga-Fuata’i (2004)

no clear linkFlanagan (2002)

moderatemoderate (C), low (I), modest (E)

Barolos (2002)

moderatemoderate (C), moderate (I),moderate (E)

modestmoderate (C), low (I), good (E)

questionableHasemann and Mansfield (1995)

ReliabilityEstimate

Validity: Criterion (C), Internal (I), External (E)

Learning withConcept Mapping

Study

Inconsistencies, Gaps & Limitations

• student mapping ability is not accounted for• there is an unsupported assumption that a

concept map represents student knowledge• validity and reliability not firmly established• not clear how they correlate with other

assessment instruments• limited studies, particularly at the secondary

and elementary levels

Part 3: Using Concept Maps in the Mathematics Classroom

Sample of Student Work

• grade 9 academic students asked to create a concept map in pairs to demonstrate how to solve a variety of single variable equations (examples of types were provided)

• topic had been discussed in class and this exercise was used as a test review

• teacher candidate (had only used concept maps for 3 days) gave a 5-10 minute demo of building a concept map explaining adding fractions

• students had not used the tool before

Example: Solving (level 1)

Example: Solving (level 2)

Example: Solving (level 3)

Example: Solving (level 4)

Approaches

• while the basic idea for constructing knowledge is to engage in student creation of maps, there are some various approaches

– map as navigator– map as a guide or as expert knowledge– partially complete map with a list of terms– partially completed map to fill in– list of words to construct a map from– list of seed terms to use or draw from– a question to resolve

Types of Maps• processes

– how do you add fractions?• organization

– what topics have we studied in this unit?– what resources do we have?

• conceptual– how do these topics relate to each other?

• problem solving– what knowledge do we have, what steps do we take?

• brainstorming– how do we approach this task?

Suggestions

• start with more structured activities (simple organizational maps may be valuable to students)

• use mapping software• let the students be creative• decide what will work in your classroom• decide what student needs you are addressing• experiment, and try the assignments on

your own first

Example: Factoring Polynomials

Example: Exponents

Example: Review

Example: Part of Large Project

Assessing Concept Maps

• assessment may be quantitative, qualitative, holistic or combinations of these, sometimes compared to an expert map

• Novak scoring system– 1 point for valid propositions and examples– 5 points for valid hierarchies– 10 points for valid crosslinks showing synthesis and 2

for less significant ones• other criteria may include directionality, accuracy,

centrality, number of links, misconceptions, structure and improvement

Activity: Area and Volume

• fill-in the concept map available at http://napaneedss.limestone.on.ca/greer/teachers/cm/measurement.ipr

• fill-in maps provide a way to add structure, focus the activity and scaffold understanding

• there are a number of possible phrases for some of the links, these may be quite indicative of understanding

• students who are comfortable can add new geometric figures

Measurement Concept Map• when complete, compare your maps…

Collaborative Exercise

1. create a map explaining how to add fractions or2. the following page contains 16 terms from a study of

first year university students• collaboratively create a concept map using these

terms, adding terms as needed• focus on linking words and phrases• making links confronts you with understanding

the relationships• discussion is encouraged in creating a map that

makes sense to all collaborators

Create a Concept Map Using…

• ratio• parallel• function• tangent• infinity• perpendicular• inverse• zero

• equation• limit• absolute value• similar• gradient• angle• variable• bisector

One Possible Representation• are there misconceptions found here?

Why Use Concept Maps?

• Will they revolutionize learning? No.• They can help

– organize– recognize misconception– represent knowledge– build knowledge

• They provide another tool for learning• They can be a change for students• They can provide an alternative

Materials Available Online

Thank you

http://napaneedss.limestone.on.ca/greer/teachers/cm/cmap.html

Gary Greergreerg@limestone.on.cahttp://napaneedss.limestone.on.ca/greer